Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.     

On a nonlinear mean and its application to image compression using multiresolution schemes

This paper is devoted to the compression of colour images using a new nonlinear cell-average multiresolution scheme. The aim is to obtain similar compression properties as linear multiresolution schemes but eliminating the classical Gibbs phenomenon of this type of reconstructions near the edges. The algorithm is based on a nonlinear reconstruction operator (using a nonlinear trigonometric mean). The new reconstruction is third-order accurate in smooth regions and adapted to the presence of discontinuities. The data used are always centred with optimal support. Some theoretical properties of this scheme are analysed (order of approximation, convergence, elimination of Gibbs effect and stability).     

Data compression based on the cubic B-spline wavelet with uniform two-scale relation

The aim of this paper is to investigate the potential artificial compression which can be achieved using an interval multiresolution analysis based on a semiorthogonal cubic B-spline wavelet. The Chui-Quak [1] spline multiresolution analysis for the finite interval has been modified [2] so as to be characterized by natural spline projection and uniform two-scale relation. Strengths and weaknesses of the semiorthogonal wavelet as regards artificial compression and data smoothing by the method of thresholding wavelet coefficients are indicated.     

On some new variational problems for image denoising

This paper is devoted to image denoising problems using multiresolution schemes related to variational problems. We start with the linear approach of Donoho and Johnstone, that is related to a well known diffusion‐type variational problem. In order to improve the behavior of this approach, we propose some new nonlinear variational problems more adapted to the problem of denoising. Moreover, the discretization is performed using nonlinear multiresolution schemes. In particular, we obtain some fast and well adapted schemes for the considered problem of denoising.     

On multiresolution schemes using a stencil selection procedure: applications to ENO schemes

This paper is devoted to multiresolution schemes that use a stencil selection procedure in order to obtain adaptation to the presence of edges in the images. Since non adapted schemes, based on a centered stencil, are less affected by the presence of texture, we propose the introduction of some weight that leads to a more frequent use of the centered stencil in regions without edges. In these regions the different stencils have similar weights and therefore the selection becomes an ill-posed problem with high risk of instabilities. In particular, numerical artifacts appear in the decompressed images. Our attention is centered in ENO schemes, but similar ideas can be developed for other multiresolution schemes. A nonlinear multiresolution scheme corresponding to a nonlinear interpolatory technique is analyzed. It is based on a modification of classical ENO schemes. As the original ENO stencil selection, our algorithm chooses the stencil within a region of smoothness of the interpolated function if the jump discontinuity is sufficiently big. The scheme is tested, allowing to compare its performances with other linear and nonlinear schemes. The algorithm gives results that are at least competitive in all the analyzed cases. The problems of the original ENO interpolation with the texture of real images seem solved in our numerical experiments. Our modified ENO multiresolution will lead to a reconstructed image free of numerical artifacts or blurred regions, obtaining similar results than WENO schemes. Similar ideas can be used in multiresolution schemes based in other stencil selection algorithms.      

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi‐banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS‐form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A Multiresolution Method for Numerical Reduction and Homogenization of Nonlinear ODEs

The multiresolution analysis (MRA) strategy for the reduction of a nonlinear differential equation is a procedure for constructing an equation directly for the coarse scale component of the solution. The MRA homogenization process is a method for building a simpler equation whose solution has the same coarse behavior as the solution to a more complex equation. We present two multiresolution reduction methods for nonlinear differential equations: a numerical procedure and an analytic method. We also discuss one possible appproach to the homogenization method.     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

Multiscale kernels

This paper reconstructs multivariate functions from scattered data by a new multiscale technique. The reconstruction uses standard methods of interpolation by positive definite reproducing kernels in Hilbert spaces. But it adopts techniques from wavelet theory and shift-invariant spaces to construct a new class of kernels as multiscale superpositions of shifts and scales of a single compactly supported function φ. This means that the advantages of scaled regular grids are used to construct the kernels, while the advantages of unrestricted scattered data interpolation are maintained after the kernels are constructed. Using such a multiscale kernel, the reconstruction method interpolates at given scattered data. No manipulations of the data (e.g., thinning or separation into subsets of certain scales) are needed. Then, the multiscale structure of the kernel allows to represent the interpolant on regular grids on all scales involved, with cheap evaluation due to the compact support of the function φ, and with a recursive evaluation technique if φ is chosen to be refinable. There also is a wavelet-like data reduction effect, if a suitable thresholding strategy is applied to the coefficients of the interpolant when represented over a scaled grid. Various numerical examples are presented, illustrating the multiresolution and data compression effects.     

A new multi-threshold image segmentation approach using state transition algorithm

Thresholding plays an important role in image segmentation and image analysis. In this paper, the normalized histogram of an image is fitted by a linear combined normal distribution functions and each normal distribution function represents a class of pixels, whereas the parameters like the mean, the variance and the weights in the fitting function are undetermined. By transforming the fitting problem into a nonlinear and non-convex optimization problem, the state transition algorithm (STA) which is a new global optimization method is used to choose the optimal parameters of the fitting function. The effectiveness of proposed approach in multilevel thresholding problems is tested by several experimental results. By comparing with OTSU, particle swarm optimization (PSO), genetic algorithm (GA) and differential evolution (DE) algorithm, it has shown that STA has competitive performance in terms of both optimization results and thresholding segmentation.     

A regularization of nonlinear diffusion equations in a multiresolution framework

We develop a regularization technique for Perona–Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill posedness of the nonlinear Perona–Malik model. The resulting algorithm is tested and the results are compared with pixel‐based methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Polyharmonic multiresolution analysis: an overview and some new results

This paper first presents a condensed state of art on multiresolution analysis using polyharmonic splines: definition and main properties of polyharmonic splines, construction of B-splines and wavelets, decomposition and reconstruction filters; properties of the so-obtained operators, convergence result and applications are given. Second this paper presents some new results on this topic: scattered data wavelet, new polyharmonic scaling functions and associated filters. Fourier transform is of extensive use to derive the tools of the various multiresolution analysis.     

A family of stable nonlinear nonseparable multiresolution schemes in 2D

Multiresolution representations of data are powerful tools in data compression. For a proper adaptation to the edges, a good strategy is to consider a nonlinear approach. Thus, one needs to control the stability of these representations. In this paper, 2D multiresolution processing algorithms that ensure this stability are introduced. A prescribed accuracy is ensured by these strategies.     

Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere

Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets. June 18, 1996. Date revised: January 14, 1997.     

Statistically based multiwavelet denoising

In this work, we consider a statistically based multiwavelet thresholding method which acts on the empirical wavelet coefficients in groups, rather than individually, in order to obtain an edge-preserving image denoising technique. Our strategy allows us to exploit the dependencies between neighboring coefficients to make a simultaneous thresholding decision, so that estimation accuracy is increased.

By interpreting the multiwavelet analysis in a statistical context, we propose a new weighted multiwavelet matrix thresholding rule, based on the statistical modeling of empirical coefficients. This allows the thresholding decision to be adapted to the local structure of the underlying image, hence producing edge-preserving denoising. Extensive numerical results are presented showing the performance of our denoising procedure.     

Learning multiresolution schemes for compression of images

We introduce a new type of multiresolution based on the Harten's framework using learning theory. This changes the point of view of the classical multiresolution analysis and it transforms an approximation problem in a learning problem opening great possibilities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The characterization of a class of subspace pseudoframes with arbitrary real number translations

In this article, the notion of generalized multiresolution structure is introduced. The concept of subspace pseudoframes with arbitrary real number translations is proposed. A new method for constructing a generalized multiresolution structure in Paley–Wiener subspace of L²(R) is presented. A pyramid decomposition scheme is established based on such a generalized multiresolution structure. Finally, affine frames of space L²(R) with arbitrary real number translations are obtained by virtue of the subspace pseudoframes and the pyramid decomposition scheme. Relation to some physical theories such as quarks confinement is also investigated.     

Hat Average Multiresolution with Error Control in 2-D

Multiresolution representations of data are a powerful tool in data compression. For a proper adaptation to the singularities, it is crucial to develop nonlinear methods which are not based on tensor product. The hat average framework permets develop adapted schemes for all types of singularities. In contrast with the wavelet framework these representations cannot be considered as a change of basis, and the stability theory requires different considerations. In this paper, non separable two-dimensional hat average multiresolution processing algorithms that ensure stability are introduced. Explicit error bounds are presented.     

Stability of nonlinear multiresolution analysis

Stability of nonlinear subdivision and multiresolution has recently been addressed in [1]. Here we give applications to convexity/monotonicity preserving schemes introduced in [2], [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Denoising using nonlinear multiscale representations

The goal of this paper is to present some numerical results for the one-dimensional denoising problem by using the nonlinear multiscale representations. We introduce modified thresholding strategies in this new context which give significant significant improvements for one-dimensional denoising problems. To cite this article: B. Matei, C. R. Acad. Sci. Paris, Ser. I 338 (2004).